1. Complex Numbers Stephanie Golmon Principia College Laser Teaching Center, Stony Brook University June 27, 2006

2. Vectors Vectors have both a magnitude and a direction. Magnitude = 20 mi Direction = 60˚ (angle of rotation from the east) *http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/vectors/u3l1a.html

3. Trigonometry The black vector is the sum of the two red vectors D = 20 mi 60˚ 20 Sin (60˚) 20 Cos (60˚) 17.32 mi 10 mi

4. 60˚ -0.50.51 -0.5 0.5 1 The Unit Circle R=1

5. Radians vs. Degrees radian: an angle of one radian on the unit circle produces an arc with arc length 1. 2 π radians = 360˚ Example: 60˚= π /3 radians *http://mathworld.wolfram.com/Radian.html

6. *http://www.math2.org/math/graphs/unitcircle.gif

7. *http://members.aol.com/williamgunther/math/ref/unitcircle.gif

8. Cosine, Sine, and the Unit Circle -0.50.51 -0.5 0.5 1 Cos(t) Sin(t)

9. Imaginary Numbers

10. Complex Numbers Have both a real and imaginary part General form: z = x +iy Z = 5 + 3i Real partImaginary part

11. Complex Plane Real axis Imaginary axis *http://www.uncwil.edu/courses/mat111hb/Izs/complex/cplane.gif

12. Vectors in the complex plane A point z=x+iy can be seen as the sum of two vectors x=Cos( θ ) y=Sin( θ ) Z=Cos( θ ) + i Sin( θ ) θ R=1 i Sin( θ) Cos( θ) z=x+iy

13. Euler’s Formula Describes any point on the unit circle θ is measured counterclockwise from the positive x, axis

14. Proof of Euler’s Formula

16. Polar Coordinates Of the form r is the distance to the point from the origin, called the modulus θ is the angle, called the argument θ = π/3 r= 20

17. Polar vs. Cartesian Coordinates Any point in the complex plane can be written in polar coordinates ( ) or in Cartesian coordinates (x+iy) how to convert between them:

18. (5+5i)(-3+3i)= ? -30 ( )( )= ? Multiplying Complex Numbers

19. (5+5i)/(-3+3i)= ? ( )/( )= ? Dividing Complex Numbers

20. Roots of Complex Numbers

21. The Most Beautiful Equation:

22. Describing Waves is the amplitude is the phase is the phase constant *derivation from: Introduction to Electrodynamics, Third Edition. David J. Griffiths. Upper Saddle River, New Jersey: Prentice Hall, 1999.

23. Continued… one period frequency angular frequency in complex notation: